Modulated filter banks have been a fundamental tool in many signal processing applications (e.g., data compression and subband adaptive filtering), mainly because they can be efficiently implemented as a polyphase digital finite impulse response (FIR) filter using the well-known discrete cosine transform (DCT) or discrete Fourier transform (DFT) for signal transformation. Modulated filter banks have been adopted in modern multimedia standards, including the Moving Picture Experts Group (MPEG) standard (see, e.g., http://www.mpeg.org), which is one of the most successful signal processing schemes in use today.
Critically sampled modulated filter banks have in general proven to be well-suited for signal compression applications in which processing of subband samples involves only quantizing coefficients, since that minimal amount of processing does not significantly increase aliasing between adjacent bands. However, for applications that require extensive subband sample processing, e.g., subband adaptive filtering (see Chapter 7 of Haykin, “Adaptive Filter Theory,” 4th edition, Prentice Hall, 2002), and subband dynamic range compression (see, Brenna, et al., “A Flexible Interbank Structure for Extensive Signal Manipulations in Digital Hearing Aids,” IEEE International Symposium on Circuits and Systems (ISCAS), vol. 6, pp. 569–572, 1998), oversampling is necessary to mitigate aliasing.
Modulated filter banks commonly operate with two windows: an analysis window and a synthesis window. The relative placement and shape of these windows determine the operation of the modulated filter bank. Portnoff, “Time-Frequency Representation of Digital Signal and Systems Based on Short-Time Fourier Analysis,” IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-28, No. 1, pp. 55–69, February 1980 (incorporated herein by reference), addressed the uniform DFT filter bank and described what is known in the art as “perfect reconstruction conditions” for it. (Those skilled in the art understand that “perfect” is a term of art, and does not mean absolute perfection in the colloquial sense.) Crochiere, “A Weighted Overlap-Add Method of Short-Time Fourier Transform,” IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-28, No. 1, pp. 99–102, February 1980, set forth a relatively efficient implementation of a DFT filter bank using a synthesis window overlap-add technique. Subsequent works (e.g., Shapiro, et al., “Design of Filters for the Discrete Short-Time Fourier Transform Synthesis,” IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 1985; Shapiro, et al., “An Algebraic Approach to Discrete Short-Time Fourier Transform Analysis and Synthesis,” ICASSP, pp. 804–807, 1984; and Bolsckei, et al., “Oversampled FIR and IIR Filter Banks and Weyl-Heisenberg Frames,” ICASSP, Vol. 3, pp. 1391–1394, 1996) address the design of the synthesis window. Although the problem of designing the synthesis window is relatively old, the techniques disclosed in these works were usually the “minimum-norm” solution. For example, Shapiro, et al., “Design of Filters . . . ,” and Shapiro, et al., “An Algebraic Approach . . . ,” both supra, proposed a minimum-norm solution for different orders of the analysis window, when the subband samples are processed, after formulating the problem as a least-square problem. In Bolsckei, et al., supra, a frame-theoretic technique was described for the design of the synthesis window using para-unitary prototypes.
Accordingly, what is needed in the art is a better technique for optimizing the operation of an oversampled DFT filter bank. What is also needed in the art is a DFT filter bank that has been optimized by the technique thereby to yield improved operation.